[Proof of Proposition REF ]
We adapt an argument in [1]}.
Let \(\mu , \eta , \nu \)  and \(\delta \)  be as given above and
let \({\varepsilon }_1 = (\delta /3)^{32}\) .
Choose \(K\)  such that
\({\mathbb {P}}[D_K \leftrightarrow \infty {\rm ~in~} G({\cal H}_\mu ) ] > 1- {\varepsilon }_1\) .
Since \({\varepsilon }_1< {\varepsilon }\)  and \(\mu < \lambda \)  this yields (i) at once.
Since \({\varepsilon }_1 < {\varepsilon }\) , we
claim that we can (and do) choose \(n_1 \in \mathbb {N}\)  with \(n_1 > K\)  such
that
\({\mathbb {P}}[U_{K,n,\lambda }] > 1- {\varepsilon }, ~~~ \forall n \in [n_1,\infty ).\) 
